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MATH
[1251]

Theorems from Infinite series

All theorems from Infinite series.

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The nn-th term test

If

lim⁡n→∞an≠0\lim_{n\to\infty}a_n\ne0

or the limit does not exist, then

∑n=1∞an\sum_{n=1}^{\infty}a_n

diverges. Equivalently,

∑an convergent⟹an→0.\sum a_n\text{ convergent}\quad\Longrightarrow\quad a_n\to0.

This is a divergence test, not a convergence test: when an→0a_n\to0, more information is needed. Apply it first, because a nonzero term limit settles the question at once.

  1. [1]:Infinite series
  2. [2]:When the terms do not vanish
  3. [3]:The harmonic series

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The nn-th term test